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Boundedness of certain oscillatory singular integrals. (English) Zbl 0886.42008
The authors prove the \(L^p(\mathbb{R}^n)\) (\(1<p<\infty\)) and \(H^1(\mathbb{R}^n)\) boundedness of oscillatory singular integral operators \(T\). These operators are defined by \[ Tf(x)= \text{p.v.} \int_{\mathbb{R}^n}e^{i\Phi(x-y)}K(x-y)f(y) dy \] where \(K(x)\) is a Calderón-Zygmund kernel and the phase function \(\Phi\in C^\infty(\mathbb{R}^n\setminus\{0\})\) is a real-valued function satisfying the conditions \[ |D^\alpha\Phi(x)|\leq C|x|^{a-|\alpha|},\quad |\alpha|\leq3, \qquad \sum_{|\alpha|=2}|D^\alpha\Phi(x)|\geq C'|x|^{a-2}, \] where \(a\in\mathbb{R}\) is some fixed number and the constants \(C, C'>0\) are independent of \(x\in\mathbb{R}^n\setminus\{0\}\) (as usual \(\alpha=(\alpha_1,\dots,\alpha_n)\in(\mathbb{N}_0)^n\), \(|\alpha|=\alpha_1+\cdots+\alpha_n\), \(D^\alpha=(\frac\partial{\partial x_1})^{\alpha_1}\cdots (\frac\partial{\partial x_n})^{\alpha_n}\)).
The results of the papers imply that for the typical example of the phase function \(\Phi(x)=|x|^a\), the \(L^p(\mathbb{R}^n)\) boundedness of \(T\) holds when \(a\neq 0\) or \(a\neq1=n\), while the \(H^1(\mathbb{R}^n)\) boundedness of \(T\) takes place when \(a\neq 0,1\). Recall that for \(a=0\) the operator \(T\) is the usual Calderón-Zygmund singular integral operator whose behaviour on the spaces \(H^1(\mathbb{R}^n)\) and \(L^p(\mathbb{R}^n)\) is well known. For \(a=n=1\) \(T\) is known to be unbounded on both \(L^p(\mathbb{R}^n)\) and \(H^1(\mathbb{R}^n)\), for \(a=1\), \(n\geq1\) the \(H^1(\mathbb{R}^n)\) boundedness of \(T\) does not hold [cf. P. Sjölin, J. Lond. Math. Soc., II. Ser. 23, 442-454 (1981; Zbl 0449.46051)].
A similar problem in the context of the Besov space \(B_0^{1,0}(\mathbb{R}^n)\) was studied earlier by one of the authors [D. Fan, “An oscillating integral in the Besov \(B^{1,0}_0(\mathbb{R}^n)\)” (to appear)].
Reviewer: P.Gurka (Praha)

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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